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Replacing the sphere measure by the ball measure in Kirchhoff's formula

Statement refuted

Statement refuted. "In the three-dimensional Kirchhoff formula one may replace the sphere measure $d S$ on $\partial B_{c t} (x)$ by Lebesgue measure $d y$ on the ball $B_{c t} (x)$ while keeping the sphere-area factor, i.e. $u (x, t) = \frac{\partial}{\partial t} [t (4 π c^{2} t^{2})^{- 1} \int_{B_{c t} (x)} u_{0}] + t (4 π c^{2} t^{2})^{- 1} \int_{B_{c t} (x)} u_{1}$ still solves the Cauchy problem; the sphere area $4 π c^{2} t^{2}$ and the ball volume are interchangeable normalisations."

Facts & Assumptions

Given: Countable Choice, $c > 0$ , and the refuted expression displayed above.

[F1]

The Kirchhoff expression's solution for constant data $(u_{0}, u_{1}) = (g_{0}, v_{0})$ is $g_{0} + t v_{0}$ ; in particular $(1, 0)$ gives $u \equiv 1$ and $(0, 1)$ gives $u = t$ (Constant initial velocity in three dimensions, Kirchhoff's formula in three dimensions).

[F2]

$∣ B_{c t} ∣ = \frac{4}{3} π c^{3} t^{3}$ and $∣ \partial B_{c t} ∣ = 4 π c^{2} t^{2}$ , since $∣ B_{r}^{3} ∣ = ω_{2} r^{3} / 3$ and $∣ \partial B_{r}^{3} ∣ = ω_{2} r^{2}$ with $ω_{2} = 4 π$ (Sphere and ball measures scale in Rn, The closed form for the volume of the unit $n$ -ball, The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ , $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[F3]

Sums, products, constant multiples of differentiable functions are differentiable with the usual rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Counterexample

1.1F1F2F3algebra

Constant displacement. Take $u_{0} \equiv 1$ , $u_{1} \equiv 0$ . The correct solution is $u \equiv 1$ by [F1], whereas replacing the surface integral by a ball integral in the actual Kirchhoff expression gives $\partial_{t} [t (4 π c^{2} t^{2})^{- 1} (4 π c^{3} t^{3} / 3)] = \partial_{t} [c t^{2} / 3] = 2 c t / 3$ . Its displacement limit is zero for every $c > 0$ , so it fails to attain $u_{0} = 1$ . It also has velocity limit $2 c / 3$ , instead of zero.

1.2F1F2F3algebra

Constant velocity. Take $u_{0} \equiv 0$ , $u_{1} \equiv 1$ . The correct solution is $u = t$ by [F1], whereas the refuted expression gives $t (4 π c^{2} t^{2})^{- 1} (4 π c^{3} t^{3} / 3) = c t^{2} / 3$ . Its velocity limit is zero, not one, and its second time derivative is $2 c / 3$ while its spatial Laplacian is zero. Thus it fails both the Cauchy data and the homogeneous wave equation for every $c > 0$ .

2.1F2step 1.1step 1.2algebra∎

The sphere-area normalisation converts a surface integral into the spherical mean. A ball integral divided by that same area instead returns $c t / 3$ times the datum when it is constant. Keeping the factors $t$ of Kirchhoff's formula then gives the two incorrect functions above. Hence sphere area and ball volume cannot be interchanged in that formula.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)